Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange equation of motion for this action is $\nabla_{\dot{\gamma}}\dot{\gamma}=0,$ where $\nabla$ is the Levi-Civita connection. This equation makes sense of course for any connection, so we may define geodesics with respect to that connection. Is there some functional on paths which depends only on the connection and yields this EOM?

share|improve this question
What kind of connection are you interested in -- how general? Just linear/affine, or something more general than that? –  Ryan Budney Oct 25 '12 at 21:02
@RyanBudney Which ones have action functionals? –  Joe Hannon Oct 26 '12 at 21:05

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.